54 ° Mr. Hell i ns's improved Solution of 
These fluents being multiplied by their proper coefficients,, 
and collected together, and their sum multiplied by the com- 
“JL 
mon factor ( a + &) % we shall have 
— 5 
2 >/(vv — cc ) , 4v /(t) y—c c ) f </(vv — cc ) 
(a -f b) 
3 c cv s 
3 c^v 
3 - 5-7 
4.6.8 
+ 
7 fl- 
ee^ 
3 5 - 7-9 
4.6.8.10 
&C. 
which fluent needs no correction, since, when v = c, the whole 
vanishes. The value of it therefore, when v = 1, will be = A V,. 
which is what we want ; and, in obtaining it, the only difficulty 
is, to compute the series in which «, £, y , &c. enter, which 
may be overcome in the manner shewn above in Art. 7. For 
the value of this series, when v = 3, will be as follows ; 
11. i 
Oi = 
** 
3 v 5 
4.6 
€ = 
3-5 V(j—cc) 
4 .6.2 
+ 
3-5 
4.6.2 
3 - 5-7 
4.6.8 
7 = 
3 - 5-7 v'(i-cc) 
4.6. 8.4 
+ 
3.57.3 cc */{ i — cc ) 
4. 6. 8. 4.2 
+ 
3 - 5 - 7-3 
4.6. 8.4. 2 
3 - 5 - 7-9 
S' _ 
. cc) 
r 
3 - 5 - 7 - 9 - 5 cc v'(i— ^') 
l 
3.57.9.5.3 cV( 1- 
CC) 
4.6.8.10 
0 — 
4.6.8.10.6 
“T 
4.6.8. 10.6.4 
- T 
4.6.8.10.6.4.2 
) SmJ iTo- 
3.5.7.9.x 1 
3.5.7.9.1 1 a/(i — cc) , 
3.5.7.9.I — cc ) , 
3.57.9.117.5^(1 
4.6.8.10.12 
£ 
"4.6.8.10.12.8 
T 
4.6.8.10.12.8.6 
r 
4.6.8.10.12.8.6.4 
& C . 
& C . 
& C . 
&c. 
Here, by attending to the same things which were observed 
in Art. 7, we may easily obtain the values of as many of the 
infinite columns of quantities on the right-hand side of the 
A 
equation as are wanted ; which, for the planets Mars, Venus„ 
the Earth, and some of the rest, are but three. 
12. The value of the series, of which a is the factor, viz ; 
4- hill _j_ 1AA- if-, &c. j, will be obtained sufficiently near 
for the purpose, by this expression, x ^ 8 ^'i^cc cty * s some “ 
what more exact than the three first terms of it ; and the infi- 
nite series 
